How many $7$-digit telephone numbers with distinct digits starting with $927$ are there?

Question

A telephone number consist of seven digits. How many phone numbers with distinct digits starting with 927 are there?

I answered $10^4$. Is that correct?

Answer 1

Since those numbers start with $927$ and the numbers are distinct you only can choose among $\{0,1,3,4,5,6,7,8\}$ distinctively $4$ more digits. Therefore the total number of such phone numbers is $$4!\binom{7}{4}=\dfrac{7!}{3!}=840$$

Answer 2

We have the first 3 digits fixed and the last 4 to be chosen among 7 (0,1,3,4,5,6,8) then, since the digits are distinct, we have

• 7 choices for the first
• 6 choices for the second
• 5 choices for the third
• 4 choices for the fourth

Now apply the Rule of Product to obtain

$$7\cdot 6 \cdot 5\cdot 4=840$$

Answer 3

Gimusi's answer is right, but just to give you a little more insight:

Whenever you need to choose $p$ elements from a set of $n$ elements, and the order you choose them matter, there are:

$\frac{n!}{(n-p)!}$ possibilities

In this problem, you have to choose 4 numbers from a set of 7, so you get:

$\frac{7!}{3!} = 840$